Problem: $\lim_{x\to \frac{\pi}{4}}\csc(x)=?$ Choose 1 answer: Choose 1 answer: (Choice A) A $\dfrac{\sqrt{2}}{2}$ (Choice B) B $\dfrac{\sqrt{3}}{2}$ (Choice C) C $\sqrt{2}$ (Choice D) D The limit doesn't exist.
Answer: $\csc(x)$ is continuous on all points in its domain. Therefore, if $x=\dfrac{\pi}{4}$ is within the domain of $\csc(x)$, we can find $\lim_{x\to \frac{\pi}{4}}\csc(x)$ by direct substitution. $x=\dfrac{\pi}{4}$ is indeed in the domain of $\csc(x)$ : $\begin{aligned} \csc\left(\dfrac{\pi}{4}\right)&=\dfrac{1}{\sin\left(\dfrac{\pi}{4}\right)} \\\\ &=\dfrac{1}{\left( \dfrac{\sqrt{2}}{2} \right)} \\\\ &=\sqrt{2} \end{aligned}$ $\lim_{x\to \frac{\pi}{4}}\csc(x)=\sqrt{2}$